Download Chapter 2 Molecular Mechanics

Chapter 3
Electronic Structures
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Introduction
• Wave function  System
• Schrödinger Equation  Wave function
– Hydrogenic atom
– Many-electron atom
• Effective nuclear charge
• Hartree product
• LCAO expansion (AO basis sets)
– Variational principle
– Self consistent field theory
– Hartree-Fock approximation
• Antisymmetric wave function
• Slater determinants
– Molecular orbital
• LCAO-MO
– Electron Correlation
Potential Energy Surfaces
• Molecular mechanics uses empirical functions for
the interaction of atoms in molecules to
approximately calculate potential energy surfaces,
which are due to the behavior of the electrons and
nuclei
• Electrons are too small and too light to be
described by classical mechanics and need to be
described by quantum mechanics
• Accurate potential energy surfaces for molecules
can be calculated using modern electronic
structure methods
Wave Function
•  is the wavefunction (contains everything we are
allowed to know about the system)
• ||2 is the probability distribution of the particles
• Analytic solutions of the wave function can be
obtained only for very simple systems, such as particle
in a box, harmonic oscillator, hydrogen atom
• Approximations are required so that molecules can be
treated
• The average value or expectation value of an operator
can be calculated by
*

 Ôd
  d
*
 O
The Schrödinger Equation
• Quantum mechanics explains how entities like electrons have both
particle-like and wave-like characteristics.
• The Schrödinger equation describes the wavefunction of a particle.
• The energy and other properties can be obtained by solving the
Schrödinger equation.
– The time-dependent Schrödinger equation
 2 2

(r , t )



V

(
r
,
t
)


i



2
m
t


– If V is not a function of time, the Schrödinger equation can be
simplified by using the separation of variables technique.
(r , t )   (r ) (t )  (t )  e iEt / 
– The time-independent Schrödinger equation
H (r )  E (r )
2 2
H
  V (r )
2m
• The various solutions of Schrödinger equation
correspond to different stationary states of the system.
The one with the lowest energy is called the ground
state.
H 1 (r )  E1 1 (r )
H 2 (r )  E2 2 (r )
H 3 (r )  E3 3 (r )
E  E  E 
1

Wave Functions at Different States
2
3
Energy
PES’s of Different States
2
1
Coordinate
3
The Variational Principle
• Any approximate wavefunction  (a trial
wavefunction) will always yield an energy
higher than the real ground state energy
E()  E when   
– Parameters in an approximate wavefunction can be
varied to minimize the Evar
– this yields a better estimate of the ground state
energy and a better approximation to the
wavefunction
– To solve the Schrödinger equation is to find the set of
parameters that minimize the energy of the resultant
wavefunction.
E()  E when   
The Molecular Hamiltonian
• For a molecular system
   (r , R)
– The Hamiltonian is made up of kinetics and potential
energy terms.
2 2
1
H
 
2m
40
e j ek
 r
j
k j
jk
Z I Z J e2 
1 
Z I e2
e2
V
 
 
 

40  I i rIi

r

R
i j i
I J I
ij
IJ 

– Atomic Units
• Length
• Charge
• Mass
• Energy
a0
e
me
hartree
o
2
a0 
 0.52917725 A
2
me e
e2
hartree 
a0
The Born-Oppenheimer Approximation
• The nuclear and electronic motions can be approximately
separated because the nuclei move very slowly with
respect to the electrons.
• The Born-Oppenheimer (BO) approximation allows the
two parts of the problem can be solved independently.
– The Electronic Hamiltonian neglecting the kinetic energy term for
the nuclei.
H
elec
Z I Z J e2
1
Z I e2
e2
2
   i  
 

2 i
I
i RI  ri
i j i ri  r j
I J  I RI  RJ
Helec  elec (r , R)  E eff ( R) elec (r , R)
– The Nuclear Hamiltonian is used for nuclear motion, describing
the vibrational, rotational, and translational states of the nuclei.
H nucl  T nucl ( R)  E eff ( R)
Nuclear motion on the BO surface
• Classical treatment of the nuclei (e,g. classical
trajectories)
 2 R nuc
E
F  ma , F  
, a
R nuc
t 2
• Quantum treatment of the nuclei (e.g. molecular
vibrations)
ˆ   
total  el  nuc , H
nuc nuc
nuc
ˆ 
H
nuc
nuclei

A
 2 2
  E (R nuc )
2m A
Solving the Schrödinger Equation
• An exact solution to the Schrödinger
equation is not possible for most of the
molecular systems.
• A number of simplifying assumptions
and procedures do make an approximate
solution possible for a large range of
molecules.
Hartree Approximation
• Assume that a many electron wavefunction can
be written as a product of one electron functions
 (r )  1 (r1 )2 (r2 )n (rn )
– If we use the variational energy, solving the many
electron Schrödinger equation is reduced to solving a
series of one electron Schrödinger equations
– Each electron interacts with the average distribution
of the other electrons
– No electron-electron interaction is accounted
explicitly
Hartree-Fock Approximation
• The Pauli principle requires that a wavefunction for
electrons must change sign when any two electrons are
permuted (1,2) = - (2,1)
• The Hartree-product wavefunction must be antisymmetrized can be done by writing the wave function
as a determinant
• Slater Determinant or HF wave function

1 (1) 1 (2)
1 2 (1) 2 (2)
1 (n)
2 (n)
n!
n (1) n (1)
n (n)
 1 2
n
Spin Orbitals
• Each spin orbital i describes the distribution of one
electron (space and spin)
• In a HF wavefunction, each electron must be in a
different spin orbital (or else the determinant is zero)
• Each spatial orbital can be combined with an alpha (, ,
spin up) or beta spin (, , spin down) component to
form a spin orbital
• Slater Determinant with Spin Orbitals
1 (r1 ) (1)
1 (r2 ) (2)
1 1 (r3 ) (3)
 (r ) 
n! 1 (r4 ) (4)
1 (r1 )  (1)
1 (r2 )  (2)
1 (r3 )  (3)
1 (r4 )  (4)
2 (r1 ) (1)
2 (r2 ) (2)
2 (r3 ) (3)
2 (r4 ) (4)
2 (r1 )  (1)   (r1 )  (1)
2 (r2 )  (2)
 (r2 )  (2)
2 (r3 )  (3)
 (r3 )  (3)
2 (r4 )  (4)
 (r4 )  (4)
n
2
n
2
n
2
n
2


1 (rn ) (n) 1 (rn )  (n) 2 (rn ) (n) 2 (rn )  (n)   (rn )  (n)
n
2
Fock Equation
• Take the Hartree-Fock wavefunction and put it
into the variational energy expression
 Ĥd


   d
*
  1 2  n
Evar
*
• Minimize the energy with respect to changes in
each orbital
Evar / i  0
yields the Fock equation
F̂i   ii
Fock Equation
• Fock equation is an 1-electron problem
F̂i   ii
– The Fock operator is an effective one electron
Hamiltonian for an orbital 
–  is the orbital energy
• Each orbital  sees the average distribution of all
the other electrons
• Finding a many electron wave function is
reduced to finding a series of one electron
orbitals
Fock Operator
ˆ V
ˆ  Jˆ  K
ˆ
Fˆ  T
NE
• kinetic energy & nuclear-electron attraction
operators
2
2
nuclei



e
ZA
2
ˆT 
V̂ne  

riA
2me
A
• Coulomb operator: electron-electron repulsion)
e2
  j rij  j d }i
electrons
Jˆ i  {

j
• Exchange operator: purely quantum mechanical arises from the fact that the wave function must
switch sign when a pair of electrons is switched
2
electrons
e
ˆ  {  
K
i d } j
i
j

rij
j
Solving the Fock Equations
F̂i   ii
1. obtain an initial guess for all the orbitals i
2. use the current i to construct a new Fock
operator
3. solve the Fock equations for a new set of i
4. if the new i are different from the old i, go
back to step 2.
When the new i are as the old i, self-consistency
has been achieved. Hence the method is also
known as self-consistent field (SCF) method
Hartree-Fock Orbitals
• For atoms, the Hartree-Fock orbitals can be
computed numerically
• The  ‘s resemble the shapes of the hydrogen
orbitals (s, p, d …)
i  R(r )Y  , 
• Radial part is somewhat different, because of
interaction with the other electrons (e.g.
electrostatic repulsion and exchange
interaction with other electrons)
Hartree-Fock Orbitals
• For homonuclear diatomic molecules, the
Hartree-Fock orbitals can also be computed
numerically (but with much more difficulty)
• the  ‘s resemble the shapes of the H2+ (1electron) orbitals
• , , bonding and anti-bonding orbitals
LCAO Approximation
• Numerical solutions for the Hartree-Fock
orbitals only practical for atoms and diatomics
• Diatomic orbitals resemble linear combinations
of atomic orbitals, e.g. sigma bond in H2
  1sA + 1sB
• For polyatomic, approximate the molecular
orbital by a linear combination of atomic
orbitals (LCAO)
   c  

Basis Function
• The molecular orbitals can be expressed as linear
combinations of a pre-defined set of one-electron
functions know as a basis functions. An
individual MO is defined
as
N
i   ci  
 1
•  : a normalized basis function
• ci : a molecular orbital expansion coefficients
    d p g p
p1
• gp : a normalized Gaussian function; g ( , r )  cx
• dp : a fixed constant within a given basis set
N
i   ci  d p g p
 1
p
n
m l r 2
y ze
The Roothann-Hall Equations
• The variational principle leads to a set of equations
describing the molecular orbital expansion coefficients,
ci, derived by Roothann and Hall
N
( F   S  )c


i
1
i
 0   1,2,  N
– Roothann Hall equation in matrix form
FC  SC
F  H    P  |    12  |  
N
N
core
 1  1
core
• H  : the energy of a single electron in the field of the bare nuclei
• P : the density matrix;
P  2
occ.orb.
 c c
*
i 1
i
i
– The Roothann-Hall equation is nonlinear and must be solved
iteratively by the procedure called the Self Consistent Field
(SCF) method.
Roothaan-Hall Equations
• Basis set expansion leads to a matrix
form of the Fock equations
FC i  SCi i




F
Ci
i
S
Fock matrix
column vector of the MO coefficients
orbital energy
overlap matrix
Solving the Roothaan-Hall Equations
1. choose a basis set
2. calculate all the one and two electron integrals
3. obtain an initial guess for all the molecular
orbital coefficients Ci
4. use the current Ci to construct a new Fock
matrix
5. solve FC i  SCi i for a new set of Ci
6. if the new Ci are different from the old Ci, go
back to step 4.
Solving the Roothaan-Hall Equations
• Known as the self consistent field (SCF) equations,
since each orbital depends on all the other orbitals,
and they are adjusted until they are all converged
• Calculating all two electron integrals is a major
bottleneck, because they are difficult (6D integrals)
and very numerous (formally N4)
• Iterative solution may be difficult to converge
• Formation of the Fock matrix in each cycle is costly,
since it involves all N4 two electron integrals
The SCF Method
• The general strategy of SCF method
– Evaluate the integrals (one- and two-electron
integrals)
– Form an initial guess for the molecular orbital
coefficients and construct the density matrix
– Form the Fock matrix
– Solve for the density matrix
– Test for convergence.
• If it fails, begin the next iteration.
• If it succeeds, proceed on the next tasks.
Closed and Open Shell Methods
• Restricted HF method (closed shell)
– Both and  electrons are forced to be in the same
N
orbital


i  i   ci  
 1
• Unrestricted HF method (open shell)
–  and  electrons are in different orbitals (different set
of ci )
N
i   ci  
 1
N
i   ci  

 1
AO Basis Sets
• Slater-type orbitals (STOs)
 n,l ,m r, ,   Nn,l ,m, Yl ,m  ,  r n1 exp  r 
• Gaussian-type orbitals (GTOs)
 a,b,c r, ,    Na' ,b,c, x a y b z c exp  r 2 
orbital radial size
• Contracted GTO (CGTOs or STO-nG)
 n,l ,m r , ,     ci  a ,b,c,i r , ,  
i
• Satisfied basis sets
• Yield predictable chemical
accuracy in the energies
• Are cost effective
• Are flexible enough to be
used for atoms in various
bonding environments
Different types of Basis
• The fundamental core & valence basis
– A minimal basis:
# CGTO = # AO
– A double zeta (DZ): # CGTO = 2 #AO
– A triple zeta (TZ):
# CGTO = 3 # AO
• Polarization Functions
– Functions of one higher angular momentum than
appears in the valence orbital space
• Diffuse Functions
– Functions with higher principle quantum number
than appears in the valence orbital space
Widely Used Basis Functions
• STO-3G: 3 primitive functions for each AO
– Single : one CGTO function for each AO
– Double : two CGTO functions for each AO
– Triple : three CGTO functions for each AO
• Pople’s basis sets
–
core
–
–
–
3-21G
valence
6-311G
Diffuse functions
6-31G**
6-311++G(d,p)
Polarization functions
Hartree Equation
• The total energy of the atomic orbital j
2 2
Ze 2
e2
 j  j 
 j  j 
 j    j (r )k (r ' )
 j (r )k (r ' )
2m
r
r  r'
k
• The LCAO expansion
   C  

j
j,
he j   j j

 he   C j ,    j     C j , 
